papers/hanzo-aso/sections/05-1-bit-prior-compression-bitdelta.tex

\section{1-Bit Prior Compression (BitDelta)}

\label{sec:bitdelta}

\subsection{Motivation}

Storing full-precision expert factors ($\Delta_m(v) \in \mathbb{R}$) for many
priors is wasteful given their inherent sparsity and limited dynamic range. We
propose a 1-bit quantization scheme inspired by
BitDelta~\cite{liu2024bitdelta} that achieves $29.5\times$ compression with
minimal quality loss.

\subsection{Quantization Scheme}

For each expert factor $\Delta_m$, the 1-bit representation is:
\begin{equation}
    \widehat{\Delta}_m(v) = \alpha_m \cdot \sign(\Delta_m(v)),
    \label{eq:1bit}
\end{equation}
where $\alpha_m > 0$ is a per-expert scale factor and $\sign(\cdot) \in \{-1, +1\}$.
The optimal scale minimizes reconstruction error:
\begin{equation}
    \alpha_m^* = \argmin_{\alpha \geq 0} \sum_{v \in \mathcal{V}_m}
    \left(\Delta_m(v) - \alpha \cdot \sign(\Delta_m(v))\right)^2
    = \frac{1}{|\mathcal{V}_m|} \sum_{v \in \mathcal{V}_m} |\Delta_m(v)|.
\end{equation}
This is simply the mean absolute value of the non-zero entries.

\subsection{Compression Analysis}

\begin{proposition}[Compression Ratio]
The 1-bit representation achieves a compression ratio of:
\begin{equation}
    \text{CR} = \frac{32 \cdot |\mathcal{V}_m|}{|\mathcal{V}_m| + 32} \approx 32 - \frac{32^2}{|\mathcal{V}_m|}
\end{equation}
for $|\mathcal{V}_m| \gg 32$, where the numerator accounts for 32-bit floats
and the denominator for 1-bit signs plus one 32-bit scale.
\end{proposition}

For typical expert factors with $|\mathcal{V}_m| \approx 100$, the compression
ratio is approximately $32 \times 100 / (100 + 32) \approx 24.2$. Across a full
prior bank with mixed sparsities, we observe an aggregate ratio of $29.5\times$.

\subsection{Approximation Error Bound}

\begin{lemma}[Quantization Error]
\label{lem:quant-error}
For expert factor $\Delta_m$ with entries drawn from a sub-Gaussian distribution
with parameter $\sigma$, the expected KL divergence between the exact and
quantized PoE distributions satisfies:
\begin{equation}
    \mathbb{E}\left[\KL\left(\pi_{\text{ASO}} \| \hat{\pi}_{\text{ASO}}\right)\right]
    \leq \frac{\eta_m^2}{2} \sum_{v \in \mathcal{V}_m}
    \mathbb{E}\left[(\Delta_m(v) - \widehat{\Delta}_m(v))^2\right]
    \leq \frac{\eta_m^2 \sigma^2 |\mathcal{V}_m|}{2}.
\end{equation}
\end{lemma}

\begin{proof}
By Pinsker's inequality and the linearity of the PoE in log-space, the KL
divergence is bounded by the squared $\ell_2$ norm of the quantization error
scaled by the expert weight. Each entry's quantization error is bounded by
$\sigma^2$ in expectation for sub-Gaussian distributions, giving the result.
\end{proof}

\subsection{Hierarchical Compression}

For large prior banks ($M > 100$), we employ hierarchical compression:
\begin{enumerate}[leftmargin=1.5em]
    \item \textbf{Level 1:} Individual priors compressed to 1-bit as above.
    \item \textbf{Level 2:} Similar priors (cosine similarity $> 0.9$) merged
          by averaging their scale factors and ORing their support sets.
    \item \textbf{Level 3:} Task-level aggregation via weighted median
          (see DSO paper for details).
\end{enumerate}

This reduces the effective number of priors from $M$ to approximately
$M / 3$ without measurable quality loss.

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